Density matrix functional theory in average and relative coordinates

Robert Erdahl, Vadene H. Smith

Ali Akbari, Norman Henry March, Ángel Rubio

The detailed nature of the correlated first-order density matrix for the model atoms in the title for arbitrary interparticle interaction $u({r}_{12})$ is studied. One representation with contracted information is first explored by constructing the momentum density $\ensuremath{\rho}(\mathbf{p})$ in terms of the wave function of the relative motion, say ${\ensuremath{\Psi}}_{R}({r}_{12})$, which naturally depends on the choice of $u({r}_{12})$. For $u({r}_{12})={e}^{2}∕{r}_{12}$, the so-called Hookean atom, and for the inverse square law $u({r}_{12})=\ensuremath{\lambda}∕{r}_{12}^{2}$, plots are presented of the above density $\ensuremath{\rho}(\mathbf{p})$ in momentum space. The correlated kinetic energy is recovered from averaging ${p}^{2}∕2m$, $m$ denoting the electron mass, with respect to $\ensuremath{\rho}(\mathbf{p})$. The second method developed is in coordinate space and expands the density matrix $\ensuremath{\gamma}({\mathbf{r}}_{1},{\mathbf{r}}_{2})$ in Legendre polynomials, using relative coordinate ${\mathbf{r}}_{1}\ensuremath{-}{\mathbf{r}}_{2}$, center-of-mass coordinate $({\mathbf{r}}_{1}+{\mathbf{r}}_{2})∕2$ and the angle, $\ensuremath{\theta}$ say, between these two vectors. For the Moshinsky atom in which $u({r}_{12})=\frac{1}{2}k{r}_{12}^{2}$ only the $s$ term $(l=0)$ contributes to the Legendre polynomial expansion. The specific example we present of the inverse square law model is shown to be characterized by the low-order terms $(s+d)$ of the Legendre expansion. The Wigner function is finally calculated analytically for both Moshinsky and inverse square law models.

J. Kluson

In this short note we study Koopman-von Neumann theory for N-particle system. We argue that it is natural to identify classical N-particle distribution function as diagonal form of density matrix operator in coordinate representation. We also determine generalized BBGKY hierarchy for reduced density matrix in coordinate representation.

Karl Blum

Fan Hong-Yi, Nian-Quan Jiang

In terms of the intermediate coordinate-momentum representation (Chin. Phys. Lett. 18 (2001) 850) and using the technique of integration within an ordered product of operators, we put the tomography theory into operator version. We reveal the new relation between the tomogram and the characteristic function of the density operator. The new expansion of the density operator in terms of the intermediate coordinate-momentum representation is also obtained.

Karl Blum